TWO RIGHT ANGLES

Book I. Propositions 13 and 14

line and makes the adjacent angles equal, then we have two right angles. But it should be obvious that when any straight line stands on another, then the adjacent angles ABC, ABD are together equal to two right angles. That is the next proposition.

(The proof will show that the two right angles CBE, EBD are equal to the three angles CBA, ABE, EBD; but angles CBA, ABD are also equal to those three angles; therefore CBA, ABD are equal to the two right angles.)

Therefore, when a straight line that stands on another straight line etc. Q.E.D.

Corollary 1. When two straight lines intersect one another, the four angles they make are together equal to four right angles.

&nbsp

Corollary 2. Therefore when any number of straight lines intersect at one point, all the angles they make are together equal to four right angles.

&nbsp

The hypothesisof Proposition 13 is that the straight line which stands on the other makes two angles. But how could it not make two angles? If it stood at the extremity of the line. In that case, it would make only one angle.

When it does not stand at the extremity, however, then the angles formed are equal to two right angles. Conversely -- if angles

ABC, ABD together are equal to two right angles, then BD is in a straight line with CB.

This is Proposition 14. But there is no previous proposition or definition that gives a criterion for two straight lines being in a straight line. This proposition is the criterion. Therefore, it can be proved only by the indirect method.

Thus, if we assume that BD is not in a straight line with CB, then we may assume that BE is, because the straight line CB may be extended in a straight line. But this leads to the conclusion that angle ABE is equal to angle ABD, the smaller to the larger; which is absurd. (Can you show that?) It follows, then, that BD is the only straight line that is in a straight line with CB.

If two straight lines are on opposite sides of a given straight line, and, meeting at one point of that line they make the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.

Let two straight lines CB, BD be on opposite sides of the straight line AF, meeting at the point B, and let the adjacent angles ABC, ABD be equal to two right angles; then BD will be in a straight line with CB.

For if BD is not in a straight line with CB, let BE be in a straight line with CB.

Then, since the straight line AB stands on the (supposed) straight line CBE,

We do not think of angles ABC, CBD together as one angle. The angle we call angle ABD is the obtuse angle ABD, which is less than two right angles.

If CD is a straight line, then, and AB meets it, then in classical

geometry we do not call CBD an angle. It has become a feature of of modern treatments, however, to call CBD a straight angle.

A straight angle is one whose sides are in a straight line with one another.

Given that concept, Proposition 14 is then both obvious and trivial. It was very likely with that in mind that a straight angle originated .

When two angles together are equal to a straight angle -- to two right angles -- we say that they are supplements of one another, or that they are supplementary angles. Thus angle ABC above is the supplement of angle ABD, and vice-versa. It will be left to Problem 5 to prove the simple theorem:

Angles that are supplements of the same angle are equal to one another.